Find the common difference if `n^(th)` term of AP is 7-4n.

Answer

To find the common difference of the arithmetic progression (AP) where the nth term is given by \( T_n = 7 - 4n \), we can follow these steps:

### Step 1: Identify the nth term and (n-1)th term
The nth term of the AP is given as:
\[ T_n = 7 - 4n \]
To find the (n-1)th term, we replace \( n \) with \( n-1 \):
\[ T_{n-1} = 7 - 4(n-1) = 7 - 4n + 4 = 11 - 4n \]

### Step 2: Calculate the common difference
The common difference \( d \) of an AP is defined as:
\[ d = T_n - T_{n-1} \]
Substituting the expressions for \( T_n \) and \( T_{n-1} \):
\[ d = (7 - 4n) - (11 - 4n) \]

### Step 3: Simplify the expression
Now, we simplify the expression:
\[ d = 7 - 4n - 11 + 4n \]
Notice that \( -4n \) and \( +4n \) cancel each other out:
\[ d = 7 - 11 = -4 \]

### Conclusion
Thus, the common difference of the arithmetic progression is:
\[ \boxed{-4} \]

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